Lecture 2 the linearization

In this lecture we go through Calderón's Born Neumann series for the forward problem and the linearization that results from truncating the higher order terms. The Fréchet derivative of the Neuman-to-Dirichlet map is derived from Calderón's calculation of the Fréchet derivative of the Dirichlet-toNeumann map with respect to conductivity.

The (high resolution) video is here Lower resolution in chunks part1, part2,part3,part4.

Some helpful resources:

• Calderón's foundational paper you can see the original paper scanned here as well as a reprint with slightly different typing errors.
• If you are confused by the difference between little-o and big-O Wikipedia may help, also definition of the Fréchet derivative
• Martin Hanke has formalised the method of finding an inclusion from Cauchy data at the boundary and describes in his paper On real-time algorithms for the location search of discontinuous conductivities with one measurement

Some homework suggestions

1. Have a go at deriving the Born-Neumann series and directly the linearization for the Neumann-to-Dirichlet case. To make it simpler assume that the conductivity does not change in a neighbourhood of the boundary
2. How does the argument change for the case of complex conductivity? Clearly you need some complex conjugates around.
   
3. Find the sensitivity/Fréchet derivative for the stationary Schrödinger/variable wave speed Helmholtz equation -∇² u + c u=0. If c is allowed to be negative you have to assume that it is "non resonant", that is you avoid eigenvalues, then there is a well defined Green's operator. This problem is relevant to ultrasound/seismic imaging, diffuse optical tomography, quantum scattering etc.